According to [1]}, a quasi-separated algebraic stack with affine stabilizers locally of finite presentation over \(\mathbb {C}\)  is lisse-étale locally linearizable over each point \(x\in \mathcal {X}\)  such that, denoting by \(G_x\)  the stabilizer subgroup of \(x\) , \(G_y\)  the stabilizer subgroup of the image \(y\)  of \(x\)  in \(\mathcal {Y}:=\mathcal {X}/T\) , and \(T_x\subset T\)  the subgroup stabilizing \(x\) , the following exact sequence is split:
\(1\rightarrow G_x\rightarrow G_y\rightarrow T_x\rightarrow 1\) 
